Abelian quotients of subgroups of the mapping class group and higher Prym representations

Abstract

A well-known conjecture asserts that the mapping class group of a surface (possibly with punctures/boundary) does not virtually surject onto $\mathbb{Z}$ if the genus of the surface is large. We prove that if this conjecture holds for some genus, then it also holds for all larger genera. We also prove that if there is a counterexample to this conjecture, then there must be a counterexample of a particularly simple form. We prove these results by relating the conjecture to a family of linear representations of the mapping class group that we call the higher Prym representations. They generalize the classical symplectic representation.

20 pages, 3 figures; appendix added containing a new counterexample in genus 1; to appear in J. London Math. Soc

References in corpus (2)

• Torelli Groups and Geometry of Moduli Spaces of Curves
• Mapping Class Groups do not have Kazhdan's Property (T)

Cited by in corpus (10)

• Quotients of surface groups and homology of finite covers via quantum representations
• Algebraic and topological properties of big mapping class groups
• Simple closed curves, finite covers of surfaces, and power subgroups of Out(F_n)
• On finiteness properties of the Johnson filtrations
• Every Infinite order mapping class has an infinite order action on the homology of some finite cover
• Homological eigenvalues of lifts of pseudo-Anosov mapping classes to finite covers
• Curves with prescribed symmetry and associated representations of mapping class groups
• An introduction to the algebraic geometry of the Putman-Wieland conjecture
• Arithmetic representations of mapping class groups
• Mapping class groups of surfaces of genus $\ge 3$ do not virtually surject to $\mathbb{Z}$